Stabilization of the Rayleigh-Taylor instability of an underlying fluid layer

In this talk we will prove that a thin layer of fluid on the ceiling is a globally stable configuration for the Muskat problem with surface tension. The initial elevation will be a medium-size perturbation of the flat state in the critical Wiener algebra $A^1$, and we will provide a new methodology which allows to perform energy estimates in Wiener algebras in the ALE formulation for the Muskat problem. Joint work with F. Gancedo & R. Granero.